Perovskites
Perovskites are a class of compounds that have a similar crystal structure to the mineral perovskite (calcium titanate—CaTiO3). Perovskites are network compounds—that is to say they are defined by a network of anions (negative ions, most commonly oxygen O2−) with specific holes for cations (positive ions, most commonly metallic ions). The most common nominal perovskite formula is ABO3, though other anions are possible and off-stoichiometry compounds with vacancies occur. The A site is a large twelve-coordinated cubo-octahedral hole typically occupied by a cation with an ionic radius greater than or equal to that of O2−, as this site is equivalent in size and coordination to an oxygen vacancy. The B site is a smaller six-coordinated octahedral hole. Ordered and disordered compounds with two (or more) cations on the A site ((AyA′(1−y))BO3) or, more commonly, the B site (A(BxB′(1−x))O3), where x and y are atomic/molar proportions with values between 0 and 1 atoms per formula unit, are possible with formula units represented by integer multiples of the simple perovskite with unit cells that can be formed by combinations of the primitive single formula unit perovskite cell.
A majority of the metals in the periodic table can be inserted into the cation sites and still have a perovskite structure as long as constraints of valence and ionic size are met. For a charge-balanced compound, the total positive charge of the cations and the negative charge of the anions must be equal. Therefore, the integer 1-1-3 ABO3 stoichiometry constrains the average cation valences on the total A and B sites of a stoichiometric oxide perovskite to be 3+ or, viewed by site, the average of the sum of the A and B valences to be 6+, i.e. Av+B(6−v)+O3, where v is an integer valence between 1 and 5. This can be accomplished with various combinations, e. g. A1+B5+O3, A2+B4+O3, A3+B3+O3.
As discussed above, complex ordered and disordered perovskite compounds with multiple A and/or B ions can occur as long as the average 3+ valence is maintained by the sum of the individual ion valences multiplied by the stoichiometry. If there is a cation vacancy fraction, that fraction is assigned a valence of zero in the average. If there is an oxygen vacancy fraction f ABO3−f then the average of the cation valences must equal the actual average number of oxygens per formula unit 3−f. If an ion of a different valence, such as a halogen, e. g. F−, substitutes on the oxygen site, then the required average valence of the cations is changed by a proportional amount. Examples of oxide perovskite cation substitutions with defined fractions include, but are not limited to:                B4+ replaced by B3+1/2B′5+1/2 (3×½+5×½=4)        B4+ replaced by B2+1/3B′5+2/3 (2×⅓+5×⅔=4)        B4+ replaced by B1+1/4B′5+3/4 (1×¼+5×¾=4)        B3+ replaced by B2+1/2B′4+1/2 (2×½+4×½=3)        A2+ replaced by A1+1/2A′3+1/2 (1×½+3×½=2)        
As is well understood by those knowledgeable in the art, real compounds can deviate slightly from these perfect stoichiometries by the presence of impurities, vacancies, interstitials, anti-site ions (e. g. A on the B-site or B on the A-site), reduced or oxidized ions and other defects without changing the fundamental nature of the compound. These deviations may commonly comprise 1% or more of the atoms or 0.05 or more of the five atomic sites in a primitive ABO3 formula unit. Further, defect compounds may have even more extensive deviations up to 10% of the atoms or 0.5 of the five atomic sites in a primitive ABO3 formula unit. Therefore, all compositions cited may be approximate, as used herein, the terms “approximate” and “approximately” indicate that the subject number can be modified by plus or minus 10% and still fall within the disclosed embodiment.
When two ions of the same valence occur on the same site, they tend to form a disordered or partially ordered continuous solid solution where the x variable can be any value from 0 to 1. An example is the well-known piezoelectric material lead zirconate titanate (PZT) with an approximate chemical formula Pb2+(Zr4+xTi4+(1−x))O3 where x varies continuously from zero to unity.
Ordered and disordered A2BB′O6 (often written as A(B1/2B′1/2)O3) compounds tend toward cubic/tetragonal/orthorhombic/monoclinic unit cells, while ordered A3BB′2O9 (often written as A(B1/3B′2/3)O3) compounds tend toward hexagonal/rhombohedral unit cells. A higher order unit cell may be constrained by symmetry to be one that is rotated with respect to a simple cubic perovskite unit cell of one ABO3 formula unit (sometimes called primitive or reduced). For example, the a and b axes of a complex unit cell are often rotated by an angle of 45° from the (1,0,0) and (0,1,0) primitive axes to the (1,1,0) and (1,−1,0) primitive axes so that the lattice parameters are increased by √{square root over (2)}. Less commonly, an axis may be rotated toward the (1,1,1) direction of the simple primitive single formula unit cell so that the lattice parameters are increased by √{square root over (3)} or to the (2,1,1) direction so the lattice parameters are increased by √{square root over (6)}. The perovskite structure is subject to polarization and distortion so the a, b and c lattice parameters may differ in tetragonal and orthorhombic crystal structures. In some of these unit cells, the lattice parameter angles may also deviate significantly from 90° in rhombohedral, hexagonal, monoclinic and triclinic structures.
The phase stability and crystallographic structure of perovskites depend on the relation between the average ionic sizes on the two sites in the crystal. The ease of formation of a perovskite for any A-B pair depends on the crystal ionic radii or, in the case of more complex structures, averaged site crystal ionic radii, and is given by the dimensionless Goldschmidt perovskite tolerance factor T:
                    T        =                                            r              A                        +                          r              O                                                          2                        ⁢                          (                                                r                  B                                +                                  r                  O                                            )                                                          (        1        )            where rA and rB are the average A- and B-site ionic radii and rO is the oxygen ionic radius commonly accepted to be 0.126 nm. It has been observed that stability of the perovskite structure may be expected within the limits T=0.88 to 1.09, but is best near unity T=1.00. The “crystal radii” as tabulated by Shannon (R. D. Shannon, “Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides,” Acta. Cryst. A32 (1976) 751) are used herein for rA and rB since they best represent the situation in the perovskite rather than the free ionic radii.
Roth has made detailed diagrams for the simple A2+-B4+ and the A3+-B3+ perovskites showing how the likely phase varies with ranges of rA and rB. See R. S. Roth, “Classification of Perovskite and Other ABO3-Type Compounds,” J. Res. N. B. S. 58 (1957) 75. Cubic perovskites occur for T closer to unity and are preferred as substrates because the properties are isotropic (especially the crystallographic and dielectric properties) and there are no anisotropic distortions, phase changes or twinning problems.
The hard sphere model can be used to estimate the cubic or pseudo-cubic hard sphere lattice parameter ahs of the perovskites as
                              a          hs                =                                                            r                A                            +                              r                O                                                    2                                +                      (                                          r                B                            +                              r                O                                      )                                              (        2        )            The values so calculated are typically higher than measured cubic/pseudo-cubic lattice parameters a by a factor that depends on the tolerance factor T and the A-site ionic radius rA. Numeric fitting gives an improved approximation for valences v=2, 3 by multiplication of ahs by this factor:a/ahs=1.102−0.247T+0.074rA  (3)Additional data and higher levels of approximation can improve this fit. Data for v=1 appear offset somewhat from this fit to a smaller factor a/ahs and even lower lattice parameters.
Crystal Growth
Perovskites comprise a number of technologically important materials because of their range of interesting properties and ability to polarize. These materials can be ferroelectric, electrooptic, ferromagnetic, ferrimagnetic, antiferromagnetic, multiferroic, piezoelectric, pyroelectric, magnetoresistive, colossal magnetoresistive (CMR), magnetooptic, photovoltaic, photoluminescent, insulating, conducting, semiconducting, superconducting, ferroelastic, catalytic, etc.
Therefore, it is important to have a variety of growth methods available to prepare single crystals, which are generally the most desired technological form. The methods that can be used to grow a crystal depend on whether the crystal composition is congruent or incongruent. Congruent compounds are those where the compound of interest crystallizes directly from a liquid of the same composition at a local maximum or minimum on the melting curve and similarly the crystal melts directly to the liquid of the same composition at a constant temperature. A congruent composition may be a stoichiometric line compound that can only have a single composition or a preferred stoichiometry of a multi-component phase that is off the nominal stoichiometric composition. The property of congruency allows the growth of large single crystals of uniform composition by bulk growth methods. Incongruent crystallization occurs when, upon cooling, a solid phase nucleates with a different composition from the liquid. Incongruent melting occurs when, upon heating, one solid phase thermodynamically transforms to an equilibrium state of another solid phase and a liquid phase each of different chemical composition from the original. Therefore, incongruent compounds are more challenging to grow and require different crystal growth methods to prepare in single crystals or films of the desired composition.
Epitaxial Growth
Epitaxy is when a crystal film of one material is grown on top of and in registry with another material, typically called a substrate. In many cases, good quality uniform incongruently melting perovskite crystals can only be grown by epitaxial means. In other cases, epitaxy is required to grow very thin films or multilayers, where the substrate provides both a template and a means of support and the thin film two-dimensional nature of the material provides important confinement or other constraints and properties for devices.
Epitaxial growth is useful for growth of materials that, for various reasons, are not amenable to growth by conventional bulk techniques wherein the crystal is grown directly from the melted target compound. The list below covers a number of cases that are relevant.                Non-congruent melting materials (including peritectic melting) may have to be grown from a melt with a composition that strongly differs from the desired crystal composition. Complex mixtures including most solid solutions have phase diagrams where the crystallizing compound can be very far off the melt composition.        High melting point oxides are often outside the limits of bulk techniques because of limits on the use temperature of the crucible and/or other furnace materials or because of a high vapor pressure of a constituent.        Compounds with volatile constituents such as lead oxide may have to be equilibrated and grown at substantially lower temperatures than their melting point to be stable. This in turn limits the solubility of the crystal constituents in a solution.        Epitaxial growth gives the ability to vary growth conditions including temperature, chemical environment and atmosphere so that unstable crystal materials can be stabilized.        
Epitaxial growth methods include, for example, liquid phase epitaxy (LPE) from 1) a high temperature solution (HTS), 2) a hydrothermal environment, 3) a partially melted flux or 4) a sol-gel. Vapor phase means include, for example, sputtering, thermal evaporation, physical vapor deposition (PVD), chemical vapor deposition (CVD), metal-organic chemical vapor deposition (MOCVD), vapor phase epitaxy (VPE), organo-metallic vapor phase epitaxy (OMVPE), molecular beam epitaxy (MBE), chemical beam epitaxy (CBE), atomic layer epitaxy (ALE), and pulsed laser deposition (PLD).
Epitaxy on a planar substrate inherently has a two-dimensional film growing in a single growth direction, though the actual growth steps may propagate laterally and there can be edge growth. Compared to point-seeded methods that require growth in three dimensions, epitaxy has a high volumetric growth rate because the crystal only needs to expand in one dimension with a planar growth front from a seed that starts with a large two-dimensional area. It is the only practical method to grow thin two-dimensional single crystal films.
Substrates
To grow quality epitaxial crystals and, in many cases, to grow the desired phase at all, the film and substrate must have an acceptable match in structure, chemistry, lattice parameter and coefficient of thermal expansion, although there are some notable exceptions. The substrate must be of good crystal quality since any defects will normally propagate into the film. It should be polished to an epitaxial (sometimes optical) quality surface with no subsurface damage and good flatness. The substrate must be chemically compatible with the film and with the epitaxial growth environment. The substrate should not contain any constituents that would fatally contaminate the film if there is interdiffusion or breakage in the melt, nor should it have any properties harmful to operation of a device containing the film, since it will, of necessity, remain part of the film/substrate composite unless it can be removed after growth (typically thick films). It should have no destructive phase transition between room temperature and the epitaxial growth temperature and preferably no phase transition at all. For a commercial technology, the substrate material should be able to be grown ≥50 mm in diameter or square dimensions. Lastly, for an economical technology, the constituents of the substrate should be of the lowest possible cost.
Substrate crystals growable by some large-scale bulk process can be matched up with non-congruent film compositions that must be grown by epitaxial techniques to produce film/substrate combinations with effective composite properties. However, there are limited choices of perovskite crystals that can be grown as uniform crystals in bulk form.
The prior art teaches that bulk crystal growth processes typically have the constraints described below.
A substrate material is most readily grown by a bulk process if it is congruently melting, i. e. having a single composition at which the liquid and solid are at equilibrium at a local maximum or minimum on the melting curve. This is necessary to produce a crystal of uniform composition. This is most typically a characteristic of a line compound of fixed or optimum stoichiometry.
Most bulk growth technologies, including the Czochralski and Bridgman methods, require that the crystal be grown from a crucible. This is typically a precious metal such as iridium or platinum, as they are inert and have high melting temperatures. The melting temperature of the substrate crystal to be grown should be within the operating range of the crucible material, typically <1600° C. for growth from platinum in a resistance or radiofrequency (RF) heated furnace and <2200° C. for growth from an iridium crucible in an RF furnace.
All substrate constituents should have as low a vapor pressure as possible under bulk growth conditions. The ionic constituents should not be easily reduced under the bulk growth conditions. If an iridium crucible is used, the material to be grown must be stable in a low oxygen atmosphere, typically ≤3% O2 and even lower for higher growth temperatures. The melted substrate material should not be reactive with or significantly dissolve the crucible material.
No destructive phase transitions should occur between the bulk growth temperature and room temperature, and preferably there should be no phase transitions at all. Perovskites are very prone to lattice distortions at room temperature, but these often do not persist to the melting temperature, where the crystal may grow in a cubic form. The easiest means to assure that the substrate crystal retains its integrity and that the grown film is uniformly in registry with the substrate is if the substrate crystal maintains the cubic structure down to room temperature.
FIG. 1 depicts a plot of known substrate materials (bottom) and some desired film materials (top) plotted on the average lattice parameter number line for the simple perovskite single formula unit ABO3 unit cell. Note that some of these materials, especially the epitaxial category, have actual unit cells that are more complex with multiple lattice parameters including integer multiples of the simple unit cell and or even higher order multiples of the lattice parameter(s) when the unit cell is rotated, but for the purposes of comparison in this plot, the lattice parameters are all converted to the simple primitive/reduced single formula unit perovskite unit cell or, in some cases, to a pseudo-cubic lattice parameter in a given plane. As can be seen from the plot, there are substrate candidates with primitive lattice parameters from 0.367 to 0.417 nm, however from 0.387 to 0.412 nm there is a dearth of cubic materials that may be readily grown. Problems with existing compounds on the number line in this range are discussed below.
The rare earth scandates DyScO3 through PrScO3 can be grown as single crystals by the Czochralski method. These materials are slightly distorted from the cubic to an orthorhombic structure, but do not have a destructive phase change and can be cut perpendicular to the c-axis to have a pseudo-cubic plane structure for substrates. However, there is anisotropic distortion in this plane in both lattice parameter and coefficient of thermal expansion. Additionally, scandium oxide is a highly refractory material and these compounds have extremely high melting temperatures close to the 2200° C. limit. Scandium is also a rare earth group metal with limited availability and high cost and the other rare earth constituents are also expensive in varying degrees and can have limited availability. These substrate materials are available for research studies from only one small quantity research lab supplier, the Leibniz Institute for Crystal Growth, and their fabrication partners. The substrates provided are small in size and high in cost. The crystal growth yield is low because the material is challenging to grow and there is a high chance of crucible failure at such melting temperatures close to the operating limit of the crucible and furnace.
Potassium tantalate KTaO3 is not congruently melting, but rather peritectic (see FIG. 2) and has been grown from an off-stoichiometry melt for scientific samples. FIG. 2 illustrates a pseudo-binary phase diagram of the system K2CO3—K2O—Ta2O5. It is referred to as pseudo-binary because there are more degrees of freedom not being considered, most specifically the oxygen stoichiometry of the perovskite phase is set at the nominal stoichiometric value of 3. The region from 75 to 100 mol % represents equilibrium between Ta2O5 and K2CO3. K2CO3 decomposes at higher temperatures and Ta2O5 concentrations, so the region from 0 to 75 mol % represents equilibrium between Ta2O5 and the decomposition product K2O. The perovskite compound of interest with approximate molar chemical formula KTaO3 melts peritectically. (A. Reisman, F. Holtzberg, M. Berkenblit, and M. Berry, ACerS-NIST Ceramic Phase Equilibria Diagrams, Version 4.0, American Ceramic Society and National Institute of Standards and Technology, (2014), Phase Diagram 00173 originally from J. Am. Chem. Soc., 78 (1956) 4514). KTaO3 may be grown by the self-flux method from an excess of K2O/K2CO3.
The principal U.S. supplier, Commercial Crystal Laboratories, has gone out of business and a few samples may be had collaboratively from Oak Ridge National Laboratory. There is at least one commercial supplier in China. The sizes available range from 5-20 mm, which is too small for any commercial product and the available commercial substrates are very expensive.
Strontium titanate SrTiO3 has a high melting point and is readily reduced unless the growth atmosphere is highly oxidizing or the growth temperature is lowered by use of a solution. Crystals are commercially prepared by flame-fusion growth and have also been grown experimentally from KF—LiF and K—Li-borate fluxes. Neither of these techniques is amenable to growth of large crystals and this material also has high costs.
Lanthanum gallate LaGaO3 and neodymium gallate NdGaO3 are not cubic and are subject to twinning.
So, for any commercial technology there is a virtual valley in the lattice parameter number line with no viable commercial congruently melting cubic perovskite substrate candidates. Therefore, innovation is required to permit growth of epitaxial crystals with a good lattice match in this lattice parameter range.
Perovskite Solid Solutions
A solid solution is a homogeneous mixture of two substances that can exist as a solid (in this case a crystalline solid) over a range of proportions between the two end members such that the crystal structure of either end member remains unchanged or only slightly distorted by the addition of the other. Even though all these materials are solids, this can be conventionally thought of as one substance (the solute) being added to another (the solvent). But, if the two end members have the same crystal structure, the two end members can fill the atomic sites of this structure by substitution, and all possible proportions between the two end members are homogeneous crystals of that structure, then this is described as a continuous solid solution. In this way, neither end member is exclusively the solvent or the solute. Solid solutions are typically thought of as disordered, whereas a perfectly ordered stoichiometric mixture is a compound with a fixed composition or range of compositions. However, varying degrees of order and disorder including clustering may be present over the range of compositions in a solid solution.
Because of the common structure, solid solutions readily form between two perovskites, where two end members, e.g., ABO3 and A′B′O3 may mix with fractions 0<x<1 to a range of ordered, partially ordered and disordered materials AxA′1−xBxB′1−xO3 (in the terminology used above, this is equivalent to x=y). If A=A′, this is somewhat simplified to ABxB′1−xO3 as is the case for some of the materials discussed in this application. Similarly, if B=B′, this is simplified to AxA′1−xBO3. However, because solid solutions are not line compounds with a fixed composition, these commonly have complex melting and crystallization behaviors as is well known to those familiar with the art. For even a simple binary solid solution phase diagram, the compound that crystallizes at any composition will, in most cases, tend to be biased toward the higher melting compound. The farther apart the two melting points, the more the liquidus (temperature where the last solid phase melts on heating or the first solid phase appears on cooling) deviates from the solidus (temperature where the last liquid phase solidifies on cooling or the first liquid phase appears on heating) and therefore more segregation occurs.
FIG. 3 illustrates this for the technologically important continuous solid solution between the perovskites potassium tantalate KTaO3 and potassium niobate KNbO3, which have lattice parameters near the region of interest ˜0.40 nm.
These two perovskites are exemplary of an ordinary continuous solid solution. (D. Rytz and H. J. Scheel, “Crystal growth of KTa1−xNbxO3 (0<x≤0.04) solid solutions by a slow-cooling method,” Journal of Crystal Growth 59 (1982) 468). In such an ordinary solid solution, the compound that crystallizes at any composition is biased toward the higher melting compound. The farther apart the two end phase melting points, the more the liquidus (heavy line) deviates from the solidus (fine line) and, therefore, more segregation occurs. As a crystal is grown from a melt of composition xL, the crystal will grow at crystallization temperature TC with a higher proportion of tantalum, xS, than in the melt, per the dashed horizontal tie line in FIG. 3. This depletes the melt of the end member KTaO3 and therefore the crystal growth path will proceed down the liquidus/solidus curves toward KNbO3 with a steadily reducing growth temperature and a steadily decreasing proportion of tantalum in the melt and in the crystal, though the solid forming at any given time will always contain more tantalum than the melt at that same time. Therefore, any crystal grown in this way will have a varying composition, properties and, if the ions are of different sizes, lattice parameter through the length and be of limited technological use.
The substrate material SAGT, Sr1.04Al0.12Ga0.35Ta0.50O3 listed in FIG. 1 is a solid solution of SAT (Sr2+Al3+1/2Ta5+1/2O3) and SGT (Sr2+Ga3+1/2Ta5+1/2O3) with two 3+ ions, Al3+ and Ga3+, on the B site. Experimentally it is observed that there is some anti-site Sr on the B site. This compound is not congruently melting and therefore has a varying lattice parameter as discussed above.
Indifferent Points
A small number of continuous solid solutions have a unique feature where the liquidus and solidus curves come together at a congruently melting maximum or minimum at some fixed proportion. This is referred to an azeotrope in vapor-liquid systems, which include the well-known ethanol-water system. For liquid-solid systems, it is properly referred to as an indifferent point and this point constitutes congruent melting. There are a substantial number of alkali halides that display this behavior, but relatively few perovskites recorded to date.
The requirements for an indifferent point to occur must be discussed in terms of the phase rule of thermodynamics. This is often oversimplified in the literature and therefore its application in this case is explained briefly here. The phase rule defines the degrees of freedom of a materials system at equilibrium F as a relation between the rank of the coefficient matrix N (equal to the number of independent components in the system) and the number of phases Π. Typically, N is taken to be equal to the number of components, but that oversimplification loses information critical to understanding indifferent points.
In this case, pressure is constant and the only variables are composition and temperature, thus the so-called “condensed” phase rule F=N−Π+1 applies. In a pseudo-binary phase diagram, it is the common teaching of the phase rule that N=2 and under these circumstances, two phases (Π=2) can come together on an equilibrium line (F=1), but an equilibrium point (F°=0) requires the meeting of three phases (Π=3). (It is referred to as pseudo-binary because there are more degrees of freedom not being considered, most specifically the oxygen stoichiometry is set at the nominal stoichiometric value of 3.) If there is a continuous solid solution, a congruently melting indifferent point (F=0) appears to violate the phase rule as there are only two phases (Π=2) a liquid and a solid of the same composition.
The problem arises in the assumption that N=2. The number of independent components is the rank of the coefficient matrix. The rank of the coefficient matrix equals the number of components if the determinant is non-zero. If the state in question strictly minimizes the total internal energy in the sense of Gibbs's minimum energy principle, then the rank of the coefficient matrix equals the number of components and Gibbs's traditional phase rule is valid (N=2 here). However, the coefficient matrix for a binary mixture can have a zero determinant at some composition, in which case the rank is 1 (same as the equations not being independent). If this is the case, the matrix form of the Gibbs-Duhem equation does not require that all differential chemical potentials be zero (dμ=0) and that the total internal energy be minimized. This is the condition that permits an indifferent point in a solution.
Another way this has been described is to view the solid solution at the indifferent point as consisting of two solid solutions with the same composition, one a solution of D in C (γ) and another of C in D (δ). If there is an ordering transition, possibly based on a combination of packing, stress and electrostatic energies, that could result in these two solid solutions being explicitly distinct.
However, the mathematical condition of a zero determinant does not usefully predict which chemical systems will have such an indifferent point. An indifferent point can occur when either the solid or liquid deviates from an ideal solution through a non-zero enthalpy of mixing (also called heat of mixing). Most commonly this occurs in the solid, so that will be discussed here, but the opposite sign of the deviation of the enthalpy of mixing in the liquid can have the same effect. The enthalpy of mixing ΔHm in a simple C-D mixture can be viewed in terms of the enthalpies of adjacent bonds between two C atoms, HCC, two D atoms, HDD and a C and a D atom, HCD.ΔHm∝x(1−x)[HCD−(HCC+HDD)/2]  (4)
If the enthalpy of the C-D bond is lower than the average of the C—C and D-D bonds, then the enthalpy of mixing is negative (more thermodynamically favorable) and it reinforces the free energy contribution of the entropy of mixing ΔSm:−tΔSm=Rt(x ln(x)+(1−x)ln(1−x))  (5)where R is the universal gas constant, t is temperature and ln is the natural logarithm. Such a negative enthalpy of mixing stabilizes the solid solution through a more negative free energy of mixing ΔGm (FIG. 4a):ΔGm=ΔHm−tΔSm  (6)A negative free energy of mixing tends to promote ordering through alternating C-D atoms and, if strong enough, will create an ordered compound at some fixed stoichiometry most likely at a congruent maximum melting temperature.
If the enthalpy of the C-D bond is higher than the average of the C—C and D-D bonds, then the enthalpy of mixing is positive (less thermodynamically favorable) and it counteracts the free energy contribution of the entropy of mixing (FIG. 4b) creating more segregation into C- and D-rich clusters and possibly an indifferent point with a minimum melting temperature. For a large positive enthalpy of mixing, the free energy curve has two local minima and complete phase separation results (FIG. 4C). A full understanding of the free energy of solidification also requires inclusion of the free energy of solidification of the two end members at the given temperature as well as the entropy of mixing of the liquid.
In FIG. 5A the classic indifferent point with minimum congruent melting is depicted where the positive enthalpy of mixing cancels out the free energy contribution of the entropy of mixing at the melting point. In this case, the differential between entropy and enthalpy contributions remains sufficiently low as the temperature is reduced further such that no spinodal decomposition is seen and there is a continuous solid solution (ss) at all temperatures. In FIG. 5B the positive enthalpy of mixing dominates at lower temperatures where the ions are still mobile and phase separation through spinodal decomposition occurs. The two phases are γ, a solid solution of D in C, and δ, a solid solution of C in D. In FIG. 5C the spinodal has risen to the melting temperature and there is no longer congruent melting, though there is complete solid solubility at the melting point. In FIG. 5D the positive heat of mixing dominates completely and phase separation into a eutectic occurs.
As above, a phase diagram with a minimum congruently melting indifferent point is depicted in FIG. 5A. When the free energy contribution of the enthalpy of mixing is greater than the contribution of the entropy of mixing, the clustering will become more extreme, resulting in the solid solution separating into two solid solutions γ (D in C) and δ (C in D) with phase separation and a eutectic, as is seen in FIGS. 5C and 5D. Because of the precise nature of the condition to form an indifferent point, it is a relatively uncommon occurrence when this condition is satisfied exactly. Also, there can still be phase separation of the solid solution below the solidus and this can result in solid phase spinodal decomposition/phase separation that is generally destructive if it occurs at a temperature where the ions are sufficiently mobile for the transformation to occur. This is depicted in FIG. 5B. Therefore, returning a congruently grown crystal intact to room temperature either requires that the decomposition temperature be below room temperature or at such a low temperature that the atoms are “frozen” in place and phase separation in any realistic time is kinetically impossible. This further constrains which systems can successfully produce crystals. The thermodynamics of compounds with multiple ions such as the oxide perovskites are more complex, but the principles are still the same.
Known Perovskite Solid Solutions with an Indifferent Point
BaTiO3—CaTiO3—
Such a congruent minimum is depicted in FIG. 6 for a continuous solid solution between the perovskites barium titanate with approximate molar chemical formula BaTiO3 and calcium titanate with approximate molar chemical formula CaTiO3.
The continuous solid solution Ba(1-x)CaxTiO3 has a cubic perovskite structure at high temperature, but the congruent composition transforms to a tetragonal phase at 98° C. most likely because the perovskite tolerance factor T=1.04. The liquidus is drawn between the points of last melting on heating (solid diamonds) and first crystallization on cooling (open diamonds). This hysteresis typically occurs from a combination of furnace hysteresis and the need for melting/crystallization to be driven by a slight degree of superheating/supercooling. The solidus is drawn from x-ray fluorescence (XRF) data on crystals grown from melts with x=0.1, 0.2 and 0.3 (crosses). (C. Kuper, R. Pankrath, and H. Hesse, Appl. Phys. A: Mater. Sci. Process, A65 (1997) 301). The phase diagram of FIG. 6 displays an indifferent point where the liquidus and solidus curves come together at a congruently melting minimum at a fixed proportion. Because of the high temperatures involved, it is impossible to characterize exactly whether or how the high temperature cubic solid solution phases on the right and left sides of the congruent point may differ.
The congruent nature of this material was verified by other authors by the growth of single crystals and determination of the uniformity of composition along the crystal length. No distinction is made by the authors between the high temperature cubic phases on the right and left sides of the congruent point. The congruent composition transforms to a tetragonal phase at 98° C. so this material is not appropriate as a substrate. This AxA′1−xBO3 solid solution has A=Ca and A′=Ba with the same valence 2+ and B=Ti in the 4+ valence state.
BaTiO3—NaNbO3—
The continuous solid solution perovskite between barium titanate with approximate molar chemical formula BaTiO3 (BT) and sodium niobate with approximate molar chemical formula NaNbO3 (NN), BT-NN, has been identified as having a similar congruently melting minimum.
FIG. 7 illustrates a pseudo-binary phase diagram of NaNbO3—BaTiO3. (D. E. Rase and R. Roy, ACerS-NIST Ceramic Phase Equilibria Diagrams, Version 4.0, American Ceramic Society and National Institute of Standards and Technology, (2014), Phase Diagram 00826 originally from Eighth Quarterly Progress Report (April 1 to Jun. 30, 1953), Vol. Appendix II, p. 16 (1953)). The continuous solid solution between the perovskites sodium niobate NaNbO3 (NN) and barium titanate BaTiO3 (BT) was first identified as having a congruently melting minimum in this reference with a minimum near 60 mole percent BaTiO3, which has a less favorable perovskite tolerance factor T=1.023.
FIG. 8 is a pseudo-binary phase diagram of the continuous solid solution NaNbO3—BaTiO3. (Merrill W. Schaefer “Phase Equilibria in the Na2O—Nb2O5 and NaNbO3—BaTiO3 Systems, and the Polymorphism of NaNbO3 and Nb2O5,” Ph.D. thesis, The Pennsylvania State University (August, 1956)). Melting solidus and liquidus curves obtained by differential thermal analysis show a congruently melting minimum in the solid solution near 50 mol percent BaTiO3, which differs from FIG. 7, and has a perovskite tolerance factor T=1.014. The phase diagram of the reference in FIG. 7 and the Schaefer thesis do not identify any phase change or phase difference in the solid solution between the two sides of the congruent point other than the BaTiO3 phase change near the end member BaTiO3 melting point, which will not impact the intermediate solid solution range of interest. No single crystals of this material were grown.
The available melting curve phase diagrams in FIG. 7 and FIG. 8 differ in the position of the congruent point between 50 and 60 mole percent BaTiO3 (x=0.5-0.6), which respectively have tolerance factors T of 1.014 and 1.023. This material has not been grown in single crystal form and nothing further about it is given in the literature beyond its initial discovery. No determination of the high temperature phase structure has been made. This material is a solid solution on both A and B sites AxA′1−xBxB′1−xO3 where A=Ba and A′=Na and B=Ti and B′=Nb have differing valences and therefore must be matched in proportion with x=y if the conditions are met that the compound is stoichiometric, all ions are single site and there are no significant vacancies or other stoichiometry defects.
LaAlO3—SrAl1/2Ta1/2O3—
A substrate material of technological importance that has this type of continuous solid solution congruent minimum is the cubic perovskite LSAT (listed in the bottom of FIG. 1 as a substrate), which is a solid solution between a simple perovskite lanthanum aluminate with approximate molar chemical formula LaAlO3 and a complex perovskite strontium aluminum tantalate with approximate molar chemical formula SrAl1/2Ta1/2O3. The composition is given as xLaAlO3-(1−x)SrAl1/2Ta1/2O3 or LaxSr1−xAl(1+x)/2Ta(1−x)/2O3, and this compound has a previously measured congruently minimum melting temperature of ˜1820° C. at x≈0.25. This is a more complex solid solution because Al is in both end members, but is not the only B ion. The proportions are constrained to a single variable x by the ionic valences and the fact that all species are single site ions. The congruent nature of this material was verified by the growth of single crystals, determination of the uniformity of composition along the crystal length and comparison to the residual melt. No determination of the high temperature phase structure has been made. This material has a reported lattice parameter of ˜0.3865 nm, below the targeted range discussed previously, and is used as a substrate for high temperature superconductors and other applications.